# Graphene - Green's function technique

## Main Question or Discussion Point

Graphene -- Green's function technique

Hi,

I am looking for a comprehensive review about using Matsubara Green's function technique for graphene (or at least some hints in the following problem). I have already learned some finite temperature Green's function technique, but only the basics.

What confuses me is that graphene has two sublattices (say A and B), and so (in principle) we have four non-interacting Green's functions: $$G_{AA}(k,\tau)=-\langle T_{\tau}a_k(\tau)a_k^{\dagger}(0)\rangle,$$ ,

where $$a_k$$ is the annihilation operator acting on the A sublattice. G_{AB}, G_{BA} and G_{BB} are defined in a similar way.

Of course, there are connections between them, but G_{AA} and G_{AB} are essentially different. Now when I am to compute e.g. the screened Coulomb potential, I do not know, which Green's function should be used to evaluate the polarization bubble.

Thank you for your help!

## Answers and Replies

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DrDu
Science Advisor

I think you will find the answer you are looking for when you consider the expression for the bubble in coordinate space.

Dear DrDu,

thank you for your response, but I do not think, I understand how your suggestion helps me. Please explain it to me a bit more thoroughly.

DrDu
Science Advisor

I mean that the electromagnetic field couples locally to the electrons. Hence the bubble is some integral containing a product of two Greensfunctions G(x,x')G(x,x'). What consequences does locality have in the case of Graphene?

Dear tejas777,

This is a very nice review, thank you very much. Let me ask just one final question: can you explain, how comes
$$F_{s,s'}(p,q)$$

in eq. (2.12) and (2.13) ?

Look at section 6.2 (on page 19/23) in:

http://nanohub.org/resources/7436/download/Notes_on_low_field_transport_in_graphene.pdf

Now, the link contains a specific example. You can probably use this type of approach to derive a more general expression, one involving the $s$ and $s'$. I may have read an actual journal article containing the rigorous analysis, but I cannot recall which one it was at the moment. If I am able to find that article I will post it here asap.